Exercise 6.3.2 (Operations between rational and irrational numbers)

Question

Prove that the sum, difference, product, and quotient of two numbers, one rational and the other a nonzero rational, are irrational.

Richard Ganaye · Accepted Answer

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

\begin{proof} 
We know that $(\Q,+,	imes)$ is a field, so $\Q$ is closed under addition, subtraction, multiplication.

Let $x \in \R \setminus \Q$ and $y \in \Q$, $y
e 0$.
\begin{itemize}
\item If $x + y \in \Q$, then $x = (x+y) - y \in \Q$. This is a contradiction, so $x + y \in \R\setminus \Q$.
\item If $x - y \in \Q$, then $x = (x-y) + y \in \Q$. This is a contradiction, so $x - y \in \R\setminus \Q$.
\item If $xy \in \Q$, since $y 
e 0$, then $x = (xy) y^{-1} \in \Q$. This is a contradiction, so $xy \in \R\setminus \Q$.
\item If $x/y \in \Q$, where $y 
e 0$, then  $x = (x/y) y \in \Q$. This is a contradiction, so $x/y \in \R\setminus \Q$.
\end{itemize}

\end{proof}

Exercise 6.3.2 (Operations between rational and irrational numbers)

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Exercise 6.3.2 (Operations between rational and irrational numbers)

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